Measuring a Building's Height With a Barometer

We contemplate a number of ways in which a humble barometer can be used to determine the height of a building, including dropping it off the top and seeing how long it takes to hit the ground.

Do you recall my recent blog about my friend, Don Wilcher, who is a lecturer at the local ITT Technical Institute here in Madison, Ala.? Well, one little tidbit of trivia I neglected to mention is that Don has invited me to give a guest lecture to his students later this month.

It probably won't surprise you to hear that the talk I'm planning to give will leap from topic to topic with the agility of a young mountain goat -- from steam engines in ancient Rome to topics that would make your eyes water, all backed up with a grab-bag of goodies for the students to look at and lay their hands on, like vacuum tubes, relays, and all sorts of other cool "stuff."

Another thing I'm planning on doing is playing one of those old "thinking outside the box" games. In fact, I was contemplating using the "old chestnut problem" that goes, "How can you use a barometer to measure the height of a tall building?"

One slight issue is that I have no clue just how much (or how little) young folks actually know these days. One problem with having a planet's worth of information at their fingertips -- via smartphones and tablets and the Internet -- is that many of our younger brethren don't actually seem to know much at all. (Maybe I'm just becoming jaded -- perhaps everyone says this about the generations that come after them.)

The bottom line is that I will make sure to commence by explaining that atmospheric pressure decreases the higher you go. Next, I will show them a barometer and explain that it is used to measure atmospheric pressure. Only then will I ask them how we might use the barometer to measure the height of a tall building.

I am, of course, expecting them to say that we could measure the atmospheric pressure at the bottom and the top of the building, and then use the difference between these two readings to determine the height of the building.

If they don't suggest this, then I fear all is lost, and we'll move on to talk about other things. But assuming they do suggest this, I will go on to explain that -- unfortunately -- there is an obscure law that forbids the use of barometers in this way, so we are obliged to come up with some other solution.

The idea is to see how many options they can come up with. I'm hoping to have anticipated all of their suggestions and to be able to amaze them with graphics that illustrate their solutions (with equations and everything!). A list of the more obvious options is as follows:

Measure the height of the barometer and then stand it next to the building when the sun is about 45 degrees in the sky. Measure the lengths of the shadows cast by the barometer and the building and then use a simple ratio to extrapolate the height of the building.

Drop the barometer off the top of the building, measure how long it takes to hit the ground, and use this value to calculate the height of the building. (I will show the simple formula and then point out that -- in order to increase the precision of our measurement -- we would have to account for things like air resistance etc.)

Hang the barometer off a long piece of string that is attached to a pole sticking out from the top of the building. Set the length of the string so that the barometer is just raised off the ground. Set the barometer swinging, and use the formula for a simple pendulum to determine the height of the building. This formula is T ~= 2.Pi times the square root of (L/g), where 'T' = the period, 'L' = the length of the string, and 'g' = the local value for gravitational acceleration. So if we measure the period and we know the value of 'g', we can calculate the length of the string 'L' (extra points will be given for the first student to point out that it would be easier to simply measure the length of the string).

The last option I know is to find the janitor who has worked in the building for years and say: "I will give you this extremely nice barometer if you tell me the height of this building."

So, this is where I need your help. Can you think of any other solutions that the students might come up with? If so, please post them as comments below. I think it's important that we show these young whippersnappers that we've "Been there, done that, read the book, seen the play, purchased the T-shirt, and even got the tattoo!"

@Duane: To keep ourselves amused while picking, we yelled back and forth between trees to collaboratively use our calculus knowledge to figure out how fast we'd be going when we hit the ground from that height.

Let's try again. first is it an aneroid or mecury barometer? If mecury then it will have a scale. Use that scale as the basis of developing a measuring stick or string. Then use the string to produce a 45 degree right angle triangle, along the top of a table and the height of the string. Then sight the top of the building along the hypotenuse of the triangle. Knowing the distance away from the building you can calculate the height of the tall building. If the building is very tall then it is easy to make both 45 and 60 degree triangles and use them to sight the top of the tall building. Now knowing the distance apart of the two sighting positions you can calculate the height of the building and the distance you are away from it.

If it is an aneroid barometer then make a simple parachute and drop the barometer attached to it from the top of the tall bulding. The barometer and parachute will quickly reach terminal velocity, so have a colleague a couple of floors of easily measurable distance down measure the time it takes to pass him/her and also the total time to reach the ground. The calculation of height is then trivial. This method avoids the problem of the calculation when throwing just the barometer off of the very tall building and a period of gravitational acceleration and then terminal velocity are involved.

Max re: "But that doesn't involve using the barometer, which is a key requirement for the exercise."

Convince the records office clerk that the barometer is actually a very valuable antique watch. Then bribe him or her with the watch to go right away and get the blueprints without delay, so you'll have enough time to stop at a barometer store and buy a new one on the way back to the building site.

Max, re: "Drop the barometer off the top of the building, measure how long it takes to hit the ground, and use this value to calculate the height of the building."

Many long years ago, I had a summer job working in the glorious Pacific Northwest forests. Part of the job involved climbing fir trees to pick the cones, which would be used in research plantings.

One day a buddy and myself were each in old growth Douglas fir trees, up in excess of 200 feet. To keep ourselves amused while picking, we yelled back and forth between trees to collaboratively use our calculus knowledge to figure out how fast we'd be going when we hit the ground from that height.